# Is best model selection by RSS equivalent to best model selection by R2 value?

I am trying to compare models using K-Fold-CV using the regsubsets function in R.

By default, it states that the ideal model is determined by the $$RSS$$.

I wished to change this parameter such that ideal model selection was performed on the basis of the $$R^2$$ value. However, I noticed that:

$$R^2 = 1 - (RSS/TSS)$$, and $$TSS = \sum \big(y_i-\bar y\big)^2$$. And therefore, is ideal model selection based on $$RSS$$ essentially equivalent to ideal model selection by $$R^2$$, given that the $$TSS$$ value is constant between test data-sets?

They are equivalent for the exact reason that you mention: one is a monotonic transformation of the other. Just keep in mind that we want to maximize $$R^2$$, while we want to minimize the residual sum of squares.
Since we typically want to find the best model for a set of data, we typically have a constant $$TSS$$.
• Note that $R^2$ can be made quite large (close to $1$) by adding many parameters that fit the noise but would not generalize. Residual sum of squares suffers from the analogous issue of being able to be made quite small (close to $0$) by using many parameters that fit to the noise in a way that would not generalize. The two approaches are, however, equivalent in the way described in the original question and in my answer.